Multiplicative function

In number theory, a multiplicative function is an arithmetic function ${\displaystyle f}$ of a positive integer ${\displaystyle n}$ with the property that ${\displaystyle f(1)=1}$ and $${\displaystyle f(ab)=f(a)f(b)}$$ whenever ${\displaystyle a}$ and ${\displaystyle b}$ are coprime.

An arithmetic function is said to be completely multiplicative (or totally multiplicative) if ${\displaystyle f(1)=1}$ and ${\displaystyle f(ab)=f(a)f(b)}$ holds for all positive integers ${\displaystyle a}$ and ${\displaystyle b}$, even when they are not coprime.

Some multiplicative functions are defined to make formulas easier to write:

• ${\displaystyle 1(n)}$: the constant function defined by ${\displaystyle 1(n)=1}$

• ${\displaystyle \operatorname {Id} (n)}$: the identity function, defined by ${\displaystyle \operatorname {Id} (n)=n}$

• ${\displaystyle \operatorname {Id} _{k}(n)}$: the power functions, defined by ${\displaystyle \operatorname {Id} _{k}(n)=n^{k}}$ for any complex number ${\displaystyle k}$. As special cases we have
  • ${\displaystyle \operatorname {Id} _{0}(n)=1(n)}$, and
  • ${\displaystyle \operatorname {Id} _{1}(n)=\operatorname {Id} (n)}$.

• ${\displaystyle \varepsilon (n)}$: the function defined by ${\displaystyle \varepsilon (n)=1}$ if ${\displaystyle n=1}$ and ${\displaystyle 0}$ otherwise; this is the unit function, so called because it is the multiplicative identity for Dirichlet convolution. Sometimes written as ${\displaystyle u(n)}$; not to be confused with ${\displaystyle \mu (n)}$.

• ${\displaystyle \lambda (n)}$: the Liouville function, ${\displaystyle \lambda (n)=(-1)^{\Omega (n)}}$, where ${\displaystyle \Omega (n)}$ is the total number of primes (counted with multiplicity) dividing ${\displaystyle n}$

The above functions are all completely multiplicative.

• ${\displaystyle 1_{C}(n)}$: the indicator function of the set ${\displaystyle C\subseteq \mathbb {Z} }$. This function is multiplicative precisely when ${\displaystyle C}$ is closed under multiplication of coprime elements. There are also other sets (not closed under multiplication) that give rise to such functions, such as the set of square-free numbers.

Other examples of multiplicative functions include many functions of importance in number theory, such as:

• ${\displaystyle \gcd(n,k)}$: the greatest common divisor of ${\displaystyle n}$ and ${\displaystyle k}$, as a function of ${\displaystyle n}$, where ${\displaystyle k}$ is a fixed integer

• ${\displaystyle \varphi (n)}$: Euler's totient function, which counts the positive integers coprime to (but not bigger than) ${\displaystyle n}$

• ${\displaystyle \mu (n)}$: the Möbius function, the parity (${\displaystyle -1}$ for odd, ${\displaystyle +1}$ for even) of the number of prime factors of square-free numbers; ${\displaystyle 0}$ if ${\displaystyle n}$ is not square-free

• ${\displaystyle \sigma _{k}(n)}$: the divisor function, which is the sum of the ${\displaystyle k}$-th powers of all the positive divisors of ${\displaystyle n}$ (where ${\displaystyle k}$ may be any complex number). As special cases we have
  • ${\displaystyle \sigma _{0}(n)=d(n)}$, the number of positive divisors of ${\displaystyle n}$,
  • ${\displaystyle \sigma _{1}(n)=\sigma (n)}$, the sum of all the positive divisors of ${\displaystyle n}$.

• ${\displaystyle \sigma _{k}^{*}(n)}$: the sum of the ${\displaystyle k}$-th powers of all unitary divisors of ${\displaystyle n}$

${\displaystyle \sigma _{k}^{*}(n)\,=\!\!\sum _{d\,\mid \,n \atop \gcd(d,\,n/d)=1}\!\!\!d^{k}}$

• ${\displaystyle \operatorname {rad} (n)}$: the radical of ${\displaystyle n}$, which is the product of the distinct prime factors of ${\displaystyle n}$.

• ${\displaystyle a(n)}$: the number of non-isomorphic abelian groups of order ${\displaystyle n}$

• ${\displaystyle \gamma (n)}$, defined by ${\displaystyle \gamma (n)=(-1)^{\omega (n)}}$, where the additive function ${\displaystyle \omega (n)}$ is the number of distinct primes dividing ${\displaystyle n}$
• ${\displaystyle \tau (n)}$: the Ramanujan tau function
• All Dirichlet characters are completely multiplicative functions, for example
  • ${\displaystyle (n/p)}$, the Legendre symbol, considered as a function of ${\displaystyle n}$ where ${\displaystyle p}$ is a fixed prime number

An example of a non-multiplicative function is the arithmetic function ${\displaystyle r_{2}(n)}$, the number of representations of ${\displaystyle n}$ as a sum of squares of two integers, positive, negative, or zero, where in counting the number of ways, reversal of order is allowed. For example:

and therefore ${\displaystyle r_{2}(1)=4\neq 1}$. This shows that the function is not multiplicative. However, ${\displaystyle r_{2}(n)/4}$ is multiplicative.

In the On-Line Encyclopedia of Integer Sequences, sequences of values of a multiplicative function have the keyword "mult".¹

See arithmetic function for some other examples of non-multiplicative functions.

A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic. Thus, if n is a product of powers of distinct primes, say n = p^a q^b ..., then f(n) = f(p^a) f(q^b) ...

This property of multiplicative functions significantly reduces the need for computation, as in the following examples for n = 144 = 2⁴ · 3²: $${\displaystyle d(144)=\sigma _{0}(144)=\sigma _{0}(2^{4})\,\sigma _{0}(3^{2})=(1^{0}+2^{0}+4^{0}+8^{0}+16^{0})(1^{0}+3^{0}+9^{0})=5\cdot 3=15}$$ $${\displaystyle \sigma (144)=\sigma _{1}(144)=\sigma _{1}(2^{4})\,\sigma _{1}(3^{2})=(1^{1}+2^{1}+4^{1}+8^{1}+16^{1})(1^{1}+3^{1}+9^{1})=31\cdot 13=403}$$ $${\displaystyle \sigma ^{*}(144)=\sigma ^{*}(2^{4})\,\sigma ^{*}(3^{2})=(1^{1}+16^{1})(1^{1}+9^{1})=17\cdot 10=170}$$

Similarly, we have: $${\displaystyle \varphi (144)=\varphi (2^{4})\,\varphi (3^{2})=8\cdot 6=48}$$

In general, if f(n) is a multiplicative function and a, b are any two positive integers, then

Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.

If f and g are two multiplicative functions, one defines a new multiplicative function ${\displaystyle f*g}$, the Dirichlet convolution of f and g, by $${\displaystyle (f\,*\,g)(n)=\sum _{d|n}f(d)\,g\left({\frac {n}{d}}\right)}$$ where the sum extends over all positive divisors d of n. With this operation, the set of all multiplicative functions turns into an abelian group; the identity element is ε. Convolution is commutative, associative, and distributive over addition.

Relations among the multiplicative functions discussed above include:

• ${\displaystyle \mu *1=\varepsilon }$ (the Möbius inversion formula)
• ${\displaystyle (\mu \operatorname {Id} _{k})*\operatorname {Id} _{k}=\varepsilon }$ (generalized Möbius inversion)
• ${\displaystyle \varphi *1=\operatorname {Id} }$
• ${\displaystyle d=1*1}$
• ${\displaystyle \sigma =\operatorname {Id} *1=\varphi *d}$
• ${\displaystyle \sigma _{k}=\operatorname {Id} _{k}*1}$
• ${\displaystyle \operatorname {Id} =\varphi *1=\sigma *\mu }$
• ${\displaystyle \operatorname {Id} _{k}=\sigma _{k}*\mu }$

The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.

The Dirichlet convolution of two multiplicative functions is again multiplicative. A proof of this fact is given by the following expansion for relatively prime ${\displaystyle a,b\in \mathbb {Z} ^{+}}$: $${\displaystyle {\begin{aligned}(f\ast g)(ab)&=\sum _{d|ab}f(d)g\left({\frac {ab}{d}}\right)\\&=\sum _{d_{1}|a}\sum _{d_{2}|b}f(d_{1}d_{2})g\left({\frac {ab}{d_{1}d_{2}}}\right)\\&=\sum _{d_{1}|a}f(d_{1})g\left({\frac {a}{d_{1}}}\right)\times \sum _{d_{2}|b}f(d_{2})g\left({\frac {b}{d_{2}}}\right)\\&=(f\ast g)(a)\cdot (f\ast g)(b).\end{aligned}}}$$

• ${\displaystyle \sum _{n\geq 1}{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}}$
• ${\displaystyle \sum _{n\geq 1}{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (s)}}}$
• ${\displaystyle \sum _{n\geq 1}{\frac {d(n)^{2}}{n^{s}}}={\frac {\zeta (s)^{4}}{\zeta (2s)}}}$
• ${\displaystyle \sum _{n\geq 1}{\frac {2^{\omega (n)}}{n^{s}}}={\frac {\zeta (s)^{2}}{\zeta (2s)}}}$

More examples are shown in the article on Dirichlet series.

An arithmetical function f is said to be a rational arithmetical function of order ${\displaystyle (r,s)}$ if there exists completely multiplicative functions g₁,...,g_r, h₁,...,h_s such that $${\displaystyle f=g_{1}\ast \cdots \ast g_{r}\ast h_{1}^{-1}\ast \cdots \ast h_{s}^{-1},}$$ where the inverses are with respect to the Dirichlet convolution. Rational arithmetical functions of order ${\displaystyle (1,1)}$ are known as totient functions, and rational arithmetical functions of order ${\displaystyle (2,0)}$ are known as quadratic functions or specially multiplicative functions. Euler's function ${\displaystyle \varphi (n)}$ is a totient function, and the divisor function ${\displaystyle \sigma _{k}(n)}$ is a quadratic function. Completely multiplicative functions are rational arithmetical functions of order ${\displaystyle (1,0)}$. Liouville's function ${\displaystyle \lambda (n)}$ is completely multiplicative. The Möbius function ${\displaystyle \mu (n)}$ is a rational arithmetical function of order ${\displaystyle (0,1)}$. By convention, the identity element ${\displaystyle \varepsilon }$ under the Dirichlet convolution is a rational arithmetical function of order ${\displaystyle (0,0)}$.

All rational arithmetical functions are multiplicative. A multiplicative function f is a rational arithmetical function of order ${\displaystyle (r,s)}$ if and only if its Bell series is of the form $${\displaystyle {\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}={\frac {(1-h_{1}(p)x)(1-h_{2}(p)x)\cdots (1-h_{s}(p)x)}{(1-g_{1}(p)x)(1-g_{2}(p)x)\cdots (1-g_{r}(p)x)}}}}$$ for all prime numbers ${\displaystyle p}$.

The concept of a rational arithmetical function originates from R. Vaidyanathaswamy (1931).

A multiplicative function ${\displaystyle f}$ is said to be specially multiplicative if there is a completely multiplicative function ${\displaystyle f_{A}}$ such that

${\displaystyle f(m)f(n)=\sum _{d\mid (m,n)}f(mn/d^{2})f_{A}(d)}$

for all positive integers ${\displaystyle m}$ and ${\displaystyle n}$, or equivalently

${\displaystyle f(mn)=\sum _{d\mid (m,n)}f(m/d)f(n/d)\mu (d)f_{A}(d)}$

for all positive integers ${\displaystyle m}$ and ${\displaystyle n}$, where ${\displaystyle \mu }$ is the Möbius function. These are known as Busche-Ramanujan identities. In 1906, E. Busche stated the identity

${\displaystyle \sigma _{k}(m)\sigma _{k}(n)=\sum _{d\mid (m,n)}\sigma _{k}(mn/d^{2})d^{k},}$

and, in 1915, S. Ramanujan gave the inverse form

${\displaystyle \sigma _{k}(mn)=\sum _{d\mid (m,n)}\sigma _{k}(m/d)\sigma _{k}(n/d)\mu (d)d^{k}}$

for ${\displaystyle k=0}$. S. Chowla gave the inverse form for general ${\displaystyle k}$ in 1929, see P. J. McCarthy (1986). The study of Busche-Ramanujan identities begun from an attempt to better understand the special cases given by Busche and Ramanujan.

It is known that quadratic functions ${\displaystyle f=g_{1}\ast g_{2}}$ satisfy the Busche-Ramanujan identities with ${\displaystyle f_{A}=g_{1}g_{2}}$. Quadratic functions are exactly the same as specially multiplicative functions. Totients satisfy a restricted Busche-Ramanujan identity. For further details, see R. Vaidyanathaswamy (1931).

Let A = F_q[X], the polynomial ring over the finite field with q elements. A is a principal ideal domain and therefore A is a unique factorization domain.

A complex-valued function ${\displaystyle \lambda }$ on A is called multiplicative if ${\displaystyle \lambda (fg)=\lambda (f)\lambda (g)}$ whenever f and g are relatively prime.

Let h be a polynomial arithmetic function (i.e. a function on set of monic polynomials over A). Its corresponding Dirichlet series is defined to be

$${\displaystyle D_{h}(s)=\sum _{f{\text{ monic}}}h(f)|f|^{-s},}$$

where for ${\displaystyle g\in A,}$ set ${\displaystyle |g|=q^{\deg(g)}}$ if ${\displaystyle g\neq 0,}$ and ${\displaystyle |g|=0}$ otherwise.

The polynomial zeta function is then

$${\displaystyle \zeta _{A}(s)=\sum _{f{\text{ monic}}}|f|^{-s}.}$$

Similar to the situation in N, every Dirichlet series of a multiplicative function h has a product representation (Euler product):

$${\displaystyle D_{h}(s)=\prod _{P}\left(\sum _{n\mathop {=} 0}^{\infty }h(P^{n})|P|^{-sn}\right),}$$

where the product runs over all monic irreducible polynomials P. For example, the product representation of the zeta function is as for the integers:

$${\displaystyle \zeta _{A}(s)=\prod _{P}(1-|P|^{-s})^{-1}.}$$

Unlike the classical zeta function, ${\displaystyle \zeta _{A}(s)}$ is a simple rational function:

$${\displaystyle \zeta _{A}(s)=\sum _{f}|f|^{-s}=\sum _{n}\sum _{\deg(f)=n}q^{-sn}=\sum _{n}(q^{n-sn})=(1-q^{1-s})^{-1}.}$$

In a similar way, If f and g are two polynomial arithmetic functions, one defines f * g, the Dirichlet convolution of f and g, by

$${\displaystyle {\begin{aligned}(f*g)(m)&=\sum _{d\mid m}f(d)g\left({\frac {m}{d}}\right)\\&=\sum _{ab=m}f(a)g(b),\end{aligned}}}$$

where the sum is over all monic divisors d of m, or equivalently over all pairs (a, b) of monic polynomials whose product is m. The identity ${\displaystyle D_{h}D_{g}=D_{h*g}}$ still holds.

Multivariate functions can be constructed using multiplicative model estimators. Where a matrix function of A is defined as $${\displaystyle D_{N}=N^{2}\times N(N+1)/2}$$

a sum can be distributed across the product$${\displaystyle y_{t}=\sum (t/T)^{1/2}u_{t}=\sum (t/T)^{1/2}G_{t}^{1/2}\epsilon _{t}}$$

For the efficient estimation of Σ(.), the following two nonparametric regressions can be considered: $${\displaystyle {\tilde {y}}_{t}^{2}={\frac {y_{t}^{2}}{g_{t}}}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(\epsilon _{t}^{2}-1),}$$

and $${\displaystyle y_{t}^{2}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(g_{t}\epsilon _{t}^{2}-1).}$$

Thus it gives an estimate value of $${\displaystyle L_{t}(\tau ;u)=\sum _{t=1}^{T}K_{h}(u-t/T){\begin{bmatrix}ln\tau +{\frac {y_{t}^{2}}{g_{t}\tau }}\end{bmatrix}}}$$

with a local likelihood function for ${\displaystyle y_{t}^{2}}$ with known ${\displaystyle g_{t}}$ and unknown ${\displaystyle \sigma ^{2}(t/T)}$.

An arithmetical function ${\displaystyle f}$ is quasimultiplicative if there exists a nonzero constant ${\displaystyle c}$ such that ${\displaystyle c\,f(mn)=f(m)f(n)}$ for all positive integers ${\displaystyle m,n}$ with ${\displaystyle (m,n)=1}$. This concept originates by Lahiri (1972).

An arithmetical function ${\displaystyle f}$ is semimultiplicative if there exists a nonzero constant ${\displaystyle c}$, a positive integer ${\displaystyle a}$ and a multiplicative function ${\displaystyle f_{m}}$ such that ${\displaystyle f(n)=cf_{m}(n/a)}$ for all positive integers ${\displaystyle n}$ (under the convention that ${\displaystyle f_{m}(x)=0}$ if ${\displaystyle x}$ is not a positive integer.) This concept is due to David Rearick (1966).

An arithmetical function ${\displaystyle f}$ is Selberg multiplicative if for each prime ${\displaystyle p}$ there exists a function ${\displaystyle f_{p}}$ on nonnegative integers with ${\displaystyle f_{p}(0)=1}$ for all but finitely many primes ${\displaystyle p}$ such that ${\displaystyle f(n)=\prod _{p}f_{p}(\nu _{p}(n))}$ for all positive integers ${\displaystyle n}$, where ${\displaystyle \nu _{p}(n)}$ is the exponent of ${\displaystyle p}$ in the canonical factorization of ${\displaystyle n}$. See Selberg (1977).

It is known that the classes of semimultiplicative and Selberg multiplicative functions coincide. They both satisfy the arithmetical identity ${\displaystyle f(m)f(n)=f((m,n))f([m,n])}$ for all positive integers ${\displaystyle m,n}$. See Haukkanen (2012).

It is well known and easy to see that multiplicative functions are quasimultiplicative functions with ${\displaystyle c=1}$ and quasimultiplicative functions are semimultiplicative functions with ${\displaystyle a=1}$.